Corollary

— If the measure $\mu$ is bounded, the relation $r < s$ implies $\mathscr{L}_F^s \subset \mathscr{L}_F^r$; moreover, the topology of convergence in mean of order $s$ is finer than the topology of convergence in mean of order $r$ (on $\mathscr{L}_F^s$).

One can show that in general the topology of convergence in mean of order $s$ is strictly finer than the topology of convergence in mean of order $r$ (Exer. 8).